Julii92
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#1
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I'm given the equation x^2\tan^2 \theta - 2ax\tan \theta + (x^2 - 2ab) = 0.
From this I'm meant to show that the maximum value of x as  \theta varies is \sqrt {a(a+2b)} and that this is achieved when  \tan \theta = \sqrt {\frac {a}{a+2b}} .

I get that the maximum value of  x occurse when  \theta = 45 , giving me the quadratic equation x^2 - ax - ab = 0 . Solving this quadratic gives me  2x = a \pm \sqrt {a(a+4b)} , which kinda looks like what I'm aiming for, but it isn't quite right. Can anyone tell me what I should be doing?

Thanks for any help.

ETA: This isn't the full question - the first part is where the projectiles element is, and all it asks is to derive the first equation given above (which wasn't a problem).
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Bric
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For the first bit, there may be better ways, but you can differentiate the expression w.r.t theta. Then as dx/dtheta = 0 at max, 3 of the terms go away, leaving a nice tan theta = a/x.

Now sub this tan theta back into the original expression to get what you need.

Then for the next bit, sub this value of x back into the tan theta = a/x bit (found earlier) and you're there.

Bric
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TenOfThem
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#3
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(Original post by Bric)
For the first bit, there may be better ways, but you can differentiate the expression w.r.t theta. Then as dx/dtheta = 0 at max, 3 of the terms go away, leaving a nice tan theta = a/x.

Now sub this tan theta back into the original expression to get what you need.

Then for the next bit, sub this value of x back into the tan theta = a/x bit (found earlier) and you're there.

Bric
You can get the tan = a/x by completing the square
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Julii92
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#4
Report Thread starter 7 years ago
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(Original post by Bric)
For the first bit, there may be better ways, but you can differentiate the expression w.r.t theta. Then as dx/dtheta = 0 at max, 3 of the terms go away, leaving a nice tan theta = a/x.

Now sub this tan theta back into the original expression to get what you need.

Then for the next bit, sub this value of x back into the tan theta = a/x bit (found earlier) and you're there.

Bric
Thanks, seems I'll have to complete C3 and C4 before tackling questions like this.
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Jammy4410
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#5
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(Original post by Julii92)
Thanks, seems I'll have to complete C3 and C4 before tackling questions like this.
On the spec, it says prior knowledge is C1/C2/C3/M1
I tried M2 but gave up after I saw trig identities.
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Julii92
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#6
Report Thread starter 7 years ago
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(Original post by Jammy4410)
On the spec, it says prior knowledge is C1/C2/C3/M1
I tried M2 but gave up after I saw trig identities.
I think it says C4 aswell - for OCR in any case.
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